# Do filters complementive to a given filter form a complete lattice? (Solved)

Let is a set. A filter (on ) is by definition a non-empty set of subsets of such that . Note that unlike some other authors I do not require . I will denote the lattice of all filters (on ) ordered by set inclusion.

Let is some (fixed) filter. Let . Obviously is a bounded lattice.

I will call complementive such filters that:

- ;
- is a complemented element of the lattice .

**Conjecture**The set of complementive filters ordered by inclusion is a complete lattice.

To this problem was found a counterexample in a MathOverflow question answer.

The counter example with a rewritten proof is also available in Filters on Posets and Generalizations online article.

## Bibliography

*Victor Porton. Do filters complementive to a given filter form a complete lattice?

* indicates original appearance(s) of problem.